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Research article

Solution-tube and existence results for fourth-order differential equations system

  • Received: 07 September 2024 Revised: 04 November 2024 Accepted: 12 November 2024 Published: 20 November 2024
  • MSC : 34A34, 34A60, 34B15, 47H10

  • In the present paper, we examine the existence of solutions to fourth-order differential equation systems when the L1-Carathéodory function is on the right-hand side. A concept of solution-tube for these issues is presented. The concepts of upper and lower solutions for fourth-order differential equations are extended to systems owing to this idea.

    Citation: Bouharket Bendouma, Fatima Zohra Ladrani, Keltoum Bouhali, Ahmed Hammoudi, Loay Alkhalifa. Solution-tube and existence results for fourth-order differential equations system[J]. AIMS Mathematics, 2024, 9(11): 32831-32848. doi: 10.3934/math.20241571

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  • In the present paper, we examine the existence of solutions to fourth-order differential equation systems when the L1-Carathéodory function is on the right-hand side. A concept of solution-tube for these issues is presented. The concepts of upper and lower solutions for fourth-order differential equations are extended to systems owing to this idea.



    In the present research, we prove existence results for a fourth-order differential equation system that takes the form:

    {ϖ(4)(t)=f(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t)),a.e. tJ=[0,1],ϖ(0)=ϖ0,ϖ(0)=ϖ1andϖ(BC), (1.1)

    where f:[0,1]×R4nRn represents an L1 -Carathéodory function, ϖ0,ϖ1Rn and (BC) can be the boundary conditions that are given by one of the following:

    (SL) Strum-Liouville boundary conditions on J

    A0ϖ(0)β0ϖ(0)=r0,A1ϖ(1)+β1ϖ(1)=r1. (1.2)

    (P) Periodic boundary conditions on J

    ϖ(0)=ϖ(1),ϖ(0)=ϖ(1), (1.3)

    where (Ai)i{0,1}Mn×n(R), such that

    i{0,1},κi0:ϖ,Aiϖκiϖ2,ϖRn
    i{0,1},riR:βi{0,1},κi+βi>0.

    We refer to [1,2,3] for further findings that were achieved in the specific instance of a boundary value issue for only one differential equation of the fourth-order (n=1), for more details, please see [4,5,6]. Existence results for higher-order differential equations can be found in [7,8], and the general case of Nth order systems is discussed in [9,10,11].

    The concept of the solution-tube of problem (1.1) is presented in this work; see [12,13,14]. This idea is inspired by [15] and [16], where solution-tubes for second and third order differential equations systems are defined, respectively, as follows:

    {ϖ(t)=f(t,ϖ(t),ϖ(t)),a.e. tJ,ϖ(BC), (1.4)

    and

    {ϖ(t)=f(t,ϖ(t),ϖ(t),ϖ(t)),a.e. tJ,ϖ(0)=ϖ0,ϖ(BC). (1.5)

    We prove that the system (1.1) has solutions. For this system, we employ the concept of a solution tube, which extends to systems the ideas of lower and upper solutions to the fourth-order differential equations presented in [17,18,19].

    The structure of this paper is given as follows: This article will utilize the notations, definitions, and findings found in Section 2. In Section 3, we provide the idea of a solution-tube to get existence results for fourth-order differential equation systems. We then go on to demonstrate the practicality of our results through two examples.

    In this section, we recall some notations, definitions, and results that we will use in this article. The scalar product and the Euclidian norm in Rn are denoted by , and , respectively. Also, let Ck(J,Rn) be the Banach space of the k-times continuously differentiable functions ϖ associated with the norm

    ϖk=max{ϖ0,ϖ0,...,ϖ(k)0},

    where

    ϖ0=max{ϖ(t):tJ}.

    The space of integral functions is denoted by L1(J,Rn), with the usual norm L1. The Sobolev space of functions in Ck1(J,Rn), where k1 and the (k1)th derivative is denoted by Wk,1(J,Rn).

    For ϖ0,ϖ1Rn, we have the following:

    Cϖ0(J,Rn):={ϖC(J,Rn):ϖ(0)=ϖ0},
    C1ϖ0,ϖ1(J,Rn):={ϖC1(J,Rn):ϖ(0)=ϖ0,ϖ(0)=ϖ1},
    CkB(J,Rn)={ϖCk(J,Rn):ϖ(BC)},
    Wk,1B(J,Rn))={ϖWk,1(J,Rn)):ϖ(BC)},
    Ck+1ϖ0,B(J,Rn)={ϖCk+1(J,Rn):ϖ(0)=ϖ0,ϖ(k)(BC)},
    Wk+1,1ϖ0,B(J,Rn))={ϖWk+1,1(J,Rn)):ϖ(0)=ϖ0,ϖ(k)(BC)},
    Ck+2ϖ0,ϖ1,B(J,Rn)={ϖCk+2(J,Rn):ϖ(0)=ϖ0,ϖ(0)=ϖ1,x(k)(BC)},
    Wk+2,1ϖ0,ϖ1,B(J,Rn))={ϖWk+2,1(J,Rn)):ϖ(0)=ϖ0,ϖ(0)=ϖ1,ϖ(k)(BC)}.

    Definition 2.1. A function f:J×R4nRn is called an L1-Carathéodory function if

    (i) For every (ϖ,y,q,p)R4n, the function tf(t,ϖ,y,q,p) is measurable;

    (ii) The function (ϖ,y,q,p)f(t,ϖ,y,q,p) is continuous for a.e. tJ;

    (iii) For every r>0, there exists a function hrL1(J,[0,)) such that f(t,ϖ,y,q,p)hr(t) for a.e. tJ and for all (ϖ,y,q,p)D, where

    D={(ϖ,y,q,p)R4n:ϖr, yr, qr, pr}.

    Definition 2.2. A function F:C3(J,Rn)×JL1(J,Rn) is integrally bounded, if for every bounded subset BC3(J,Rn), there exists an integral function hBL1(J,[0,)) so that F(ϖ,α)(t)hB(t), for tJ,(ϖ,α)B×J.

    The operator NF:C3(J,Rn)×JC0(J,Rn) will be associated with F and defined by

    NF(ϖ)(t)=t0F(ϖ,α)(s)ds.

    We now state the following results:

    Theorem 2.1. [20] Let F:C3(J,Rn)×JL1(J,Rn) be continuous and integrally bounded, then NF is continuous and completely continuous.

    Lemma 2.1. [21] Let E be a Banach space. Let v:JE be an absolutely continuous function, then for

    {tJ:v(t)=0andv(t)0},

    the measure is zero.

    Lemma 2.2. [22] For wW2,1(J;R) and ε0, assume that one of the next properties is satisfied:

    (i) w(t)εw(t)0; for almost every tJ,κ0w(0)ν0w(0)0,κ1w(1)+ν1w(1)0; where κi,νi0,max{κi,νi}>0;i=0,1;andmax{κ0,κ1,ε}>0,

    (ii) w(t)εw(t)0; for almost every tJ,ε>0,w(0)=w(1), w(1)w(0)0,

    (iii) w(t)εw(t)0; for almost every t[0,t1][t2,1],ε>0,w(0)=w(1), w(1)w(0)0, w(t)0,t[t1,t2].

    Then w(t)0, t[0,1].

    Lemma 2.3. [22] Let fC(J×R2n,Rn) be a L1-Carathéodory function (see definition in [22]). Consider the following problem:

    {ϖ(t)=f(t,ϖ(t),ϖ(t)),a.e. tJ,ϖ(BC). (2.1)

    Let ε>0, and (z,N) a solution-tube of (2.1) given in Definition 2.3 of [22]. If ϖW2,1B(J,Rn) satisfies

    Π(t)=ϖ(t)z(t),ϖ(t)z(t)+ϖ(t)z(t)2ϖ(t)z(t)ϖ(t)z(t),ϖ(t)z(t)2ϖ(t)z(t)3εϖ(t)z(t)N(t)εN(t),

    a.e. on

    {tJ:ϖ(t)z(t)>N(t)}.

    Then

    ϖ(t)z(t)N(t) for every tJ.

    Now, we recall some properties of the Leray Schauder degree. The interested reader can see [23,24].

    Theorem 2.2. Let E be a Banach space and UE is an open bounded set. We define KU(¯U,E)={f:¯UE, where f is compact and f(ϖ)ϖ, for every ϖU}, the Leary-Schauder degree on U of (Idf) is an integer deg(Idf,U,0) satisfying the following properties:

    (i) (Existence) If deg(Idf,U,0)0, then ϖU, s.t.,

    ϖf(ϖ)=0.

    (ii) (Normalization) If 0U, then deg(Id,U,0)=1.

    (iii) (Homotopy invariance) If h:¯U×JE is a compact such that ϖh(ϖ,α)0 for each (ϖ,α)U×J, then

    deg(Idh(.,α),U,0)=deg(Idh(.,0),U,0), for every αJ.

    (iv) (Excision) If VU is open and ϖf(ϖ)0 for all ϖ¯U V, then

    deg(Idf,U,0)=deg(Idf,V,0).

    (v) (Additivity) If U1,U2U are disjoint and open, such that ¯U=¯U1U2 and ϖf(ϖ)0 for all ϖU1U2, then

    deg(Idf,U,0)=deg(Idf,U1,0)+deg(Idf,U2,0).

    In this section, we define the solution-tube to the problem (1.1). This definition is important for our discussion about the existence results. A solution to this problem is a function ϖW4,1(J,Rn) satisfying (1.1). Now, we define the tube solution of problem (1.1), where the functions zW4,1(J,Rn) and NW4,1(J,[0,) are chosen before studying the existence of this problem.

    Definition 3.1. Let (z,N)W4,1(J,Rn)×W4,1(J,[0,)). The couple (z,N) is solution-tube of (1.1), if

    (i) N(t)0,tJ.

    (ii) For almost every tJ and for all (ϖ,y,q,p)F,

    qz(t),f(,t,ϖ,y,q)z(t)+pz(t)2N(t)N4(t)+(N(t))2,

    where

    F={(ϖ,y,q,p)R4n:ϖz(t)N(t),yz(t)N(t),qz(t)=N(t),qz(t),pz(t)=N(t)N(t)}.

    (iii) z(4)(t)=f(t,ϖ,y,z(t),z(t)),a.e.t[0,1] such that N(t)=0 and (ϖ,y)R2n, such that ϖz(t)N(t) and yz(t)N(t).

    (iv) With (1.2), we have

    r0(A0z(0)β0z(0))κ0N(0)β0N(0),
    r1(A1z(1)+β1z(1))κ1N(1)+β1N(1).

    If (BC) is given by (1.3), then

    z(0)=z(1), N(0)=N(1),z(1)z(0)N(1)N(0).

    (v) ϖ0z(0)N(0), ϖ1z(0)N(0).

    The next notation will be used

    T(z,N)={ϖC2(J, Rn):ϖ(t)z(t)N(t),ϖ(t)z(t)N(t)andϖ(t)z(t)N(t)foralltJ}.

    The next hypotheses will be used:

    (F1) f:J×R4nRn is a L1-Carathéodory function.

    (H1) There exists (z,N)W4,1(J,Rn)×W4,1(J,[0,)) a solution-tube of the main system (1.1).

    The next family of problems should be considered to prove the general existence theorem that will be presented:

    {ϖ(4)(t)εϖ(t)=fεα(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t)),a.e. tJ,ϖ(0)=ϖ0, ϖ(0)=ϖ1andϖ(BC), (3.1)

    where ε,αJ and fεα:J×R4nRn is defined by

    fεα(t,ϖ,y,q,p)={α(N(t)qz(t)f1(t,ϖ,y,˜q,˘p)ε˜q)ε(1α)z(t)+(1αN(t)qz(t))(z(4)(t)+N(4)(t)qz(t)(qz(t))),ifqz(t)>N(t),α(f1(t,ϖ,y,q,p)εq)ε(1α)z(t)+(1α)(z(4)(t)+N(4)(t)N(t)(qz(t))),otherwise,

    where (z,N) is the solution-tube of (1.1),

    f1(t,ϖ,y,q,p)={f(t,ˉϖ,ˆy,q,p),    if ϖz(t)>N(t) and yz(t)>N(t),f(t,ϖ,y,q,p),    otherwise,
    ˉϖ(t)=N(t)ϖz(t)(ϖz(t))+z(t), (3.2)
    ˆy(t)=N(t)yz(t)(yz(t))+z(t), (3.3)
    ˜q(t)=N(t)qz(t)(qz(t))+z(t), (3.4)
    ˘p(t)=p+(N(t)qz(t),pz(t)qz(t))(qz(t)qz(t)), (3.5)

    and where we mean

    N(4)(t)N(t)(qz(t))=0on{tJ:q(t)z(t)=N(t)=0}.

    We associate with fεα the operator Fε:C3(J,Rn)×JL1(J,Rn) defined by

    Fε(ϖ,α)(t)=fεα(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t)).

    Similarly to the Lemma 3.3 and Propositions 3.4 in [20] and results in [25], we need the following auxiliary results:

    Lemma 3.1. Assume (H1). If a function ϖW4,1ϖ0,ϖ1,B(J,Rn) satisfies

    ϖ(t)z(4)(t)+ϖ(t)z(t)2ϖ(t)z(t)ϖ(t)z(t),ϖ(t)z(t)2ϖ(t)z(t)3εϖ(t)z(t)N(4)(t)εN(t),

    for a.e. t{tJ:ϖ(t)z(t)>N(t)}, then ϖT(z,N).

    Proof. By assumption

    ϖW3,1ϖ1,B(J,Rn), ϖW2,1B(J,Rn),

    and thus, from applying Lemma 2.3 to ϖ, we obtain

    ϖ(t)z(t)N(t),tJ.

    On

    {tJ:ϖ(t)z(t)>N(t), ϖ(t)z(t)ϖ(t)z(t)N(t).}

    The function

    tϖ(t)z(t)N(t),

    is nonincreasing on J. Since

    ϖ0z(0)N(0),

    we get

    ϖ(t)z(t)N(t),tJ,

    hence

    ϖ(t)z(t)ϖ(t)z(t)N(t).

    The function

    tϖ(t)z(t)N(t),

    is nonincreasing on J and since

    ϖ(0)z(0)N(0),

    we obtain

    ϖ(t)z(t)N(t),tJ.

    Proposition 3.1. Assume (F1) and (H1) hold. Then the operator Fε that was defined earlier is continuous and integrally bounded.

    Proof. First, we will prove that Fε is integrally bounded. If ϖB, where B is a bounded set of C3(J,Rn), K>0 that satisfies ϖ(i)(t)K, tJ, where i=0,1,2,3. Then fεα(t,.,.,.,.) is bounded in E, it can be observed that

    Fε(ϖ,α)(t)=fεα(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))max{f(t,ϖ,y,q,p),(ϖ,y,q,p)E}+|N(t)|+z(t)+z(4)(t)+|N(4)(t)|,

    for all αJ and almost every tJ, where

    E={(u,y,q,p)R4n:uz0+N0,yz0+N0,qz0+N0,p2ϖ0+z0+N0}.

    As f is L1-Carathéodory, zW4,1(J,Rn) and NW4,1(J,[0,)), it is easy to see that Fε is integrally bounded.

    In order prove the continuity, we should firstly prove that if (ϖp,αp)(ϖ,α) in C3(J,Rn)×J, then

    fεαp(t,ϖp(t),ϖp(t),ϖp(t),ϖp(t))fεα(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))a.e. tJ. (3.6)

    Using the fact that f is L1-Carathéodory, and from the definition of fεα, it can be concluded that (3.6) is true a.e. on {tJ:ϖ(t)z(t)N(t)}. Then, by Lemma 2.1 and Proposition 3.5 in [22], we easily show that ˘ϖn(t)ϖ(t) on

    {tJ:ϖ(t)z(t)=N(t)>0},

    where ˘ϖn(t), is defined as (3.5). Then, (3.6) is satisfied on

    {tJ:ϖ(t)z(t)=N(t)>0}.

    For

    A={tJ:ϖ(t)z(t)=N(t)=0},

    where ϖ(t)=z(t), and by Lemma 2.1, it is not hard to see that ϖ(t)=z(t), N(t)=0 and N(4)(t)=0, tA, which means,

    fεα(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))=α(f1(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))εϖ(t))+(1α)(z(4)(t)εz(t))=αf1(t,ϖ(t),ϖ(t),z(t),z(t))εz(t)+(1α)z(4)(t),

    a.e. on A. By the solution tube hypothesis (Definition 3.1 condition (iii)), we have

    fεα(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))=αz(4)(t)+(1α)z(4)(t)εz(t)=z(4)(t)εz(t),

    a.e. on A. Consequently, (3.6) must be true a.e. on J. Using the Lebesgue-dominated convergence theorem, and since Fε is integrally bounded, the proof can be concluded.

    Now, we can obtain our general existence result. We follow the method of proof given in [20].

    Theorem 3.1. Assume (F1), (H1), and the following conditions are satisfied:

    (Hk) For every solution ϖ of the related system (3.1), K>0, so that

    ϖ(t)<K,tJ.

    Then, problem (1.1) has a solution ϖW4,1(J,Rn)T(z,N).

    Proof. We first show that if (ϖ,N)W4,1ϖ0,ϖ1,B(J,Rn)×W4,1(J,[0,)) is a solution of (3.1), then

    ϖ(t)z(t)N(t),tJ.

    For the set

    {tJ:ϖ(t)z(t)>N(t)}.

    By the definition of ~ϖ and ˘ϖ(t) (as (3.4) and (3.5)), we have

    ~ϖ(t)z(t)=N(t), (3.7)
    <~ϖ(t)z(t),˘ϖ(t)z(t)>=N(t)N(t).

    Also

    ˘ϖ(t)z(t)2=ϖ(t)z(t)2+(N(t))2ϖ(t)z(t),ϖ(t)z(t)2ϖ(t)z(t)2.

    Then, by (H1), we obtain

    ϖ(t)z(t)z(4)(t)+ϖ(t)z(t)2ϖ(t)z(t)ϖ(t)z(t),ϖ(t)z(t)2ϖ(t)z(t)3εϖ(t)z(t)=ϖ(t)z(t),fεα(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))+εϖ(t)ϖ(t)z(t)+1ϖ(t)z(t)(ϖ(t)z(t)2ϖ(t)z(t),ϖ(t)z(t)2ϖ(t)z(t)2)εϖ(t)z(t)=ϖ(t)z(t),αN(t)ϖ(t)z(t)(f1(t,ϖ(t),ϖ(t),~ϖ(t),˘ϖ(t))z(4)(t))ϖ(t)z(t)+ϖ(t)z(t),(1αN(t)ϖ(t)z(t))N(4)(t)(ϖ(t)z(t))ϖ(t)z(t)ϖ(t)z(t)εϖ(t)z(t),α(~ϖ(t)z(t))(ϖ(t)z(t))ϖ(t)z(t)+˘ϖ(t)z(t)2(N(t))2ϖ(t)z(t)εϖ(t)z(t)=αϖ(t)z(t)~ϖ(t)z(t),f1(t,ϖ(t),ϖ(t),~ϖ(t),˘ϖ(t))z(4)(t)+N(4)(t)(1αN(t)ϖ(t)z(t))εϖ(t)z(t)+εϖ(t)z(t)εϖ(t)z(t),α(~ϖ(t)z(t))ϖ(t)z(t)+˘ϖ(t)z(t)2(N(t))2ϖ(t)z(t)αϖ(t)z(t)(N(t)+(N(t))2˘ϖ(t)z(t)2)+N(4)(t)αN(t)ϖ(t)z(t)αεN(t)+˘ϖ(t)z(t)2(N(t))2ϖ(t)z(t)=N(4)(t)αεN(t)+(1α)(˘ϖ(t)z(t)2(N(t))2)ϖ(t)z(t)N(4)(t)εN(t),

    on

    {tJ:ϖ(t)z(t)>N(t)}.

    Using Lemma 3.1, it can be observed that any solutions to system (3.1) are in T(z,N) and then, in U, where

    U={ϖC3(J,Rn):u(i)0z(i)0+N(i)0+1,i=1,0,2;ϖ0K}.

    Fix εJ such that the operator Lε:C1B(J,Rn)C0(J,Rn) given by

    Lε(ϖ)(t)=ϖ(t)ϖ(0)εt0ϖ(s)ds

    is invertible.

    Consider the linear operator D:C3ϖ0,ϖ1,B(J,Rn)C1B(J,Rn) defined by

    D(ϖ)=ϖ.

    It can be easily confirmed that D is invertible.

    A solution to (1.1) is a fixed point of the operator

    K=D1oL1εoNFε:C3(J,Rn)×JC3ϖ0,ϖ1,B(J,Rn)C3(J,Rn).

    Using Proposition 3.1 and Theorem 2.1, and since the operators D and Lε are continuous, it can be concluded that K is completely continuous and fixed point free on U. Let

    K0:C3(J,Rn)×JC3(J, Rn)

    by K0(ϖ,α)=αK(ϖ,0). Because Fε(.,0) is integrally bounded, there exists an open bounded set KC3(J,Rn), where

    UK and K0(C3(J,Rn)×J)K,

    it can be implied from the homotopic and the excision properties of the Leray-Schauder theorem that

    1=deg(Id,K,0)=°(IdK0(.,1),K,0)=deg(IdK(.,0),K,0)=°(IdK(.,0),U,0)=deg(IdK(.,1),U,0).

    As a result, there exists a solution ϖT(z,N) for α=1 to (3.1), which also can solve (1.1) by definition of fε1. The proof is complete.

    Now, following from our general existence theorem (Theorem 3.1), other existence results will be presented. We will consider the following assumptions:

    (H2) There exist a function γL1(J,[0,)) and a Borel measurable function ΨC([0,),[1,)) s.t.

    (ⅰ) f(t,ϖ,y,q,p)γ(t)Ψ(p),tJ and (ϖ,y,q,p)R4n, where ϖz(t)N(t), yz(t)N(t) and qz(t)N(t),

    (ⅱ) c0, we have

    cdτΨ(τ)=.

    (H3) There exist, a function γL1(J,[0,)) and a Borel measurable function ΨC([0,],]0,)) s.t.

    (ⅰ) p,f(t,ϖ,y,q,p)Ψ(p)(γ(t)+p),tJ and (ϖ,y,q,p)R4n, where ϖz(t)N(t), yz(t)N(t) and qz(t)N(t),

    (ⅱ) c0, we have

    cτdτΨ(τ)+τ=.

    (H4) r,b>0, c0 and a function hL1(J,R) s.t. tJ, (ϖ,y,q,p)R4n, where

    ϖz(t)N(t),yz(t)N(t),qz(t)N(t),

    and pr, then

    (b+cq)σ(t,ϖ,y,q,p)ph(t),

    where

    σ(t,ϖ,y,q,p)=q,f(t,ϖ,y,q,p)+p2pp,f(t,ϖ,y,q,p)q,pp3.

    (H5) a0 and lL1(J,R) s.t.

    f(t,ϖ,y,q,p)a(q,f(t,ϖ,y,q,p)+p2)+l(t),

    tJ and (ϖ,y,q,p)R4n, where

    ϖz(t)N(t),yz(t)N(t),

    and

    qz(t)N(t).

    Theorem 3.2. Assume (F1), (H1), and (\mathcal{H}2) aresatisfied.If (BC) isgivenby (1.2) with \max \left\{ \beta _{0}, \beta _{1}\right\} > $$ 0 $, then system $ (1.1) $ has at least one solution $ \varpi \in T(z, N)\cap W^{4,1}( \mathcal{J}, \mathbb{\ R}^{n}) $.

    Proof. Theorem 3.1 will guarantee the existence of a solution if we can obtain a priori bound on the third derivative of any solution ϖ to (3.1). It is known that ϖT(z,N) from the Theorem 3.1 proof. Therefore, since (BC) is given by (1.2) with max{β0,β1}>0, k>0, s.t.

    min{ϖ(0),ϖ(1)}k.

    Now, let R>k such that

    RkdsΨ(s)>L=γL1+εN0+z(4)L1+N(4)L1.

    Suppose there exists t1[0,1] s.t. ϖ(t1)R. Then, there exists t0t1[0,1] such that ϖ(t0)=k and ϖ(t)k, t[t0,t1]. Let us assume that t0<t1. Thus, by (H2), almost everywhere on [t0,t1], we have

    ϖ(t)=ϖ(t)ϖ(t)ϖ(4)(t)f(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))+εϖ(t)z(t)+z(4)(t)+|N(4)(t)|γ(t)Ψ(ϖ(t))+εN(t)0+z(4)(t)+N(4)L1.

    So,

    t1t0ϖ(t)tΨ(ϖ(t))dtL.

    Then, we have

    t1t0ϖ(t)tΨ(ϖ(t))dt=ϖ(t1)ϖ(t0)dsΨ(s)RkdsΨ(s)>L,

    which contradict the assumptions. So, for any solution ϖ of (3.1), R>0 s.t. ϖ(t)<R, tJ.

    If (H2) is replaced by (H3), extra assumptions are needed.

    Theorem 3.3. Assume (F1), (H1), (H3), and (\mathcal{H}4) or (\mathcal{H}5) aresatisfied.Then,thereexistsasolution \varpi \in T(z, N)\cap W^{4,1}(\mathcal{J}, \mathbb{\ R}^{n}) to (1.1) $.

    For this end, we need the next three Lemmas.

    Lemma 3.2. [20] Let r,k0, NL1([0,1],R) and ΨC([0,[,]0,[) be a Borel measurable function s.t.

    rτdτΨ(τ)>NL1([0,1],R)+k.

    Then K>0, s.t. ϖ0<K, ϖW2,1([0,1],Rn) satisfy:

    (i) mint[0,1]ϖ(t)r;

    (ii) ϖL1([t0,t1],R)k for every interval [t0,t1]{t[0,1]:ϖ(t)r};

    (iii) |ϖ(t),ϖ(t)|Ψ(ϖ(t))(N(t)+ϖ(t)) a.e. on

    {t[0,1]:ϖ(t)r}.

    Lemma 3.3. [20] Let r,ν>0, γ0 and NL1([0,1],R). Then there exists a nondecreasing function ωC[0,[,[0,[) s.t.

    ϖL1([t0,t1],R)ω(ϖ0),

    and

    mint[0,1]ϖ(t)max{r,;ω(ϖ0)}.

    uW2,1([0,1],Rn) and

    {t[t0,t1]:ϖ(t)r},

    the following inequality

    (ν+γϖ(t))σ0(t,ϖ)+γϖ(t),ϖ(t)2ϖ(t)ϖ(t)ϖ(t)N(t)

    is satisfied, where

    σ0(t,ϖ)=ϖ(t),ϖ(t)+ϖ2ϖ(t)ϖ(t),ϖ(t)ϖ(t),ϖ(t)ϖ(t)3.

    Lemma 3.4. [20] Let K>0, and NL1([0,1],R). Then there exists an increasing function ωC([0,[,]0,[) s.t. ϖL1([0,1],R)ω(ϖ0) for all ϖW2,1([0,1],Rn) that satisfies

    ϖ(t)k(ϖ(t),ϖ(t)+ϖ(t)2)+N(t),

    for almost every t[0,1].

    Proof of Theorem 3.3. Similarly to the previous proof, we need Theorem 3.1 to prove that the third derivative of all solutions ϖ to (3.1) is bounded. Let ϖ be a solution to (3.1), where ϖT(z,N) from Theorem 3.1 proof. We obtain from (H3),

    |ϖ(t),ϖ(4)(t)||ϖ(t),f(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))|+(εϖ(t)z(t)+z(4)(t)+|N(4)(t)|)ϖ(t)(γ(t)+ϖ(t))Ψ(ϖ(t))+(ε|N(t)|+z(4)(t)+|N(4)(t)|)ϖ(t)(Ψ(ϖ(t))+ϖ(t))+(γ(t)+ϖ(t)+ε|N(t)|+z(4)(t)+|N(4)(t)|),

    for almost every t[0,1]. Thus, condition (iii) of Lemma 3.2 is verified, where

    ψ(τ)=Ψ(τ)+τandN(τ)=γ(τ)+ε|N(τ)|+z(4)(τ)+|N(4)(t)|.

    Therefore, it is enough to prove that conditions (i) and (ii) are verified. (H4) guarantees that a.e. on

    {t[0,1]:ϖ(t)r},

    we have

    σ0(t,ϖ)=ϖ(t)+ϖ(t)2ϖ(t)ϖ(t)ϖ(t),ϖ(t)ϖ(t)3=ασ(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))+(1α)ϖ(t)     +(1α)ϖ(t)+(ε+N(4)(t)N(t))(ϖ(t)z(t))ϖ(t)     (1α)ϖ(t)+(ε+N(4)(t)N(t))(ϖ(t)z(t))ϖ(t),ϖ(t)ϖ(t)3ασ(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))+(1α)ϖ(t)    2(z(t)+|N(t)|)(z(4)(t)+ε|N(t)|+|N(4)(t)|)r.

    Thus, we have

    (b+cϖ(t))σ0(t,ϖ)+cϖ(t),ϖ(t)2ϖ(t)ϖ(t)αϖ(t)+b(1α)ϖ(t)h(t)δ0,(t),

    where

    δ0(t)=2r(b+cz(t)+c|N(t)|)(z(t)+|N(t)|)(z(4)(t)+ε|N(t)|+|N(4)(t)|).

    If we take

    z=minα[0,1]{α+b(1α)},ν=bzandθ=cη,

    we can apply Lemma 3.3 to ϖ([0,1],Rn). Thus, conditions of Lemma 3.2 are verified. Moreover, if (H5) holds, we have

    ϖ(4)(t)αf(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))+εϖ(t)z(t)+z(4)(t)+|N(4)(t)|αa(ϖ(t),f(t,ϖ(t),ϖ(t),ϖ(t),ϖ(t))+ϖ(t)2)+l(t)        +ε|N(t)|+z(4)(t)+|N(4)(t)|a(ϖ(t),ϖ(4)(t)+ϖ(t)2)+ε|N(t)|+z(4)(t)+|N(4)(t)|        a(1α)ϖ(t),z(4)(t)+(N(4)(t)N(t)+ε)(ϖ(t)z(t))a(ϖ(t),ϖ(4)(t)+ϖ(t)2)+ε|N(t)|+z(4)(t)+|N(4)(t)|        +a(z(t)+|N(t)|)(z(4)(t)+|N(4)(t)|+εN(t)).

    Therefore, if Lemma 3.4 is applied to ϖ([0,1],Rn), all conditions of Lemma 3.2 are satisfied. As a result, for all solutions ϖ of (3.1), ϖ0<K for some constant K>0.

    From the previous results, we obtain the following consequence:

    Corollary 3.1. Assume (F1), (H1), (H2), and (H4) or (H5) are satisfied. Then, we have a solution ϖT(z,N)W4,1([0,1],Rn) to the system (1.1).

    Remark 3.1. Definition 3.1 is associated to the definitions of lower and upper solutions to the fourth-order differential equation. These definitions are used in [17], and introduce them for problems (1.1) and (1.2).

    Definition 3.2. Let n=1 and ϖ0=ϖ1=0.

    A function κC4(]0,1[)C3(J) is called a lower solution to (1.1) and (1.2), if

    (i) κ(4)(t)f(t,κ(t),κ(t),κ(t),κ(t)) for every tJ;

    (ii) κ(0)=κ(0)=0;

    (iii) A0κ(0)β0κ(0)r0 and A1κ(1)+β1κ(1)r1.

    On the other hand, an upper solution to (1.1) and (1.2) is a function νC4(]0,1[)C3(J) that satisfies (i)–(iii) with reversed inequalities.

    Similarly to Remark 3.2 in [20], we consider the following assumptions:

    (A) There exist lower and upper solutions, κ and ν, respectively, to (1.1) and (1.2), where κν .

    (B) There exists a solution-tube (z,N) to (1.1) and (1.2).

    (C) There exist lower and upper solutions, κν, to (1.1) and (1.2) s.t.

    (i) κ(t)ν(t)) for all tJ;

    (ii) f(t,ν(t),ν(t),q,p)f(t,ϖ,y,q,p)f(t,κ(t),κ(t),q,p); tJ and (ϖ,y,q,p)R4n such that κ(t)ϖν(t) and κ(t)ϖ(t)ν(t).

    It can be easily checked that

    If (B) holds with z and N of class C4, and z(0)=N(0)=0, then (A) holds.

    Indeed, κ=zN and ν=z+N are respectively lower and upper solutions of (1.1) and (1.2). However, (A) does not imply (B).

    Noting that (B) is more general than (C), see [17]; i.e.,

    If (C) is verified, then (B) is verified.

    Taking z=(κ+ν)/2 and N=(νκ)/2. But, (B) does not imply (C) (ii) and κ(0)=ν(0)=0.

    Next, we present two examples to illustrate the applicability of Theorem 3.3.

    Example 3.1. Consider the following system:

    {ϖ(4)(t)=ϖ(t)+ϖ(t)(ϖ(t)2ϖ(t)ϖ(t),ϖ(t)ϖ(t))ξ,a.e. tJ,ϖ(0)=0,ϖ(0)=0,A0ϖ(0)=0,A1ϖ(1)+βt=1ϖ(1)=0, (3.8)

    here ξRn,ξ=1, and Aiandβi are given before for i=0,1. Show that when z0,N(t)=t36, (z,N) is a solution-tube of (3.8). We have (H3) and (H4) are verified for

    Ψ(τ)=3τ+1,γ(t)=0,b=1,c=0,r>0,h(τ)=2τr+τ5.

    Owing to the Theorem 3.3, the problem (3.8) has at least one solution ϖ s.t.

    ϖ(t)t36, ϖ(t)t22 and ϖ(t)t for all tJ.

    Example 3.2. Consider the following system:

    {ϖ(4)(t)=ϖ(t)(ϖ(t)2+1)+φ(t),a.e. tJ,ϖ(0)=0,ϖ(0)=0,ϖ(0)=ϖ(1),ϖ(0)=ϖ(1), (3.9)

    where φL(J,Rn) with φL1. Show that for z0,N(t)=t22, (z,N) is a solution-tube of (3.9). We have (H3) and (H5) are verified when

    Ψ(τ)=τ2+2, γ(t)=0,a=1,l(t)=3.

    By Theorem 3.3, the problem (3.9) has at least one solution ϖ s.t.

    ϖ(t)t22, ϖ(t)t, ϖ(t)1,tJ.

    Our paper discusses the existence of solutions for fourth-order differential equation systems, focusing particularly on cases involving L1-Carathéodory functions on the right-hand side of the equations. We first, introduced the concept of a solution-tube, which is an innovative approach that extends the concepts of upper and lower solutions applicable to fourth-order equations into the domain of systems. It outlines the mathematical framework necessary to demonstrate that solutions exist for these types of differential equation systems under specified boundary conditions (such as Sturm-Liouville and periodic conditions). The paper stands on prior results regarding higher-order differential equations, providing a fresh perspective and methodology that can be used to explore further developments in the field. In addition to presenting the theoretical underpinnings, we also illustrated the practicality of our results with examples, contributing to the mathematical discourse on differential equations and our solutions, which ultimately serves as a scholarly contribution to understanding the dynamics of fourth-order systems and the existence of their solutions; please see [26,27].

    Bouharket Bendouma: Conceptualization, formal analysis, Writing-original draft preparation; Fatima Zohra Ladrani and Keltoum Bouhali: investigation, Methodology; Ahmed Hammoudi and Loay Alkhalifa: Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

    The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

    The authors declare that there is no conflict of interest.



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